Regular Grammars

To be a right-linear grammar, every production of the grammar must have one of the two forms V T*V or V T*.

To be a left-linear grammar, every production of the grammar must have one of the two forms V VT* or V T*.

You do not get to mix the two. For example, consider a grammar with the following productions:

S

S a X

X S b

This grammar is neither right-linear nor left-linear, hence it is not a regular grammar. We have no reason to suppose that the language it generates is a regular language (one that is generated by a dfa).

In fact, the grammar generates a language whose strings are of the form ab. This language cannot be recognized by a dfa. (Why not?)